# How can l prove that Newton's laws are time invariant?

• stefano77
In summary, Newton's law can be proven to be time invariant by showing that if x(t) is a solution of dd/ddx x(t) = f(x(t)), then y(t) = x(-t) is also a solution of dd/ddt y(t) = f(y(t)). This can be further demonstrated by noting that the second derivative with respect to time for y(t) is equal to the second derivative with respect to time of x(-t), which is equivalent to f(x(-t)).
stefano77
Misplaced Homework Thread -- Moved to the Schoolwork forums by the Mentors
how can l prove Newton's law is time invariant?

if x (t) is a solution of dd/ddx x(t) = f(x(t)) then if l put y(-t) dd/ddt y(t)=dd/ddt x(-t). Now how dd/ddt x(-t) is equal to f(x(-t))?dd/ddt is second derivative with respect to time

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topsquark and vanhees71
stefano77 said:
how can l prove Newton's law is time invariant?

if x (t) is a solution of dd/ddx x(t) = f(x(t)) then if l put y(-t) such that dd/ddt y(t)=dd/ddt x(-t). Now how dd/ddt x(-t) is equal to f(x(-t))?dd/ddt is second derivative with respect to time
##\dfrac{d^2}{dt^2} y(-t) = \dfrac{d^2}{dt^2} x(t)##

or equivalently
##\dfrac{d^2}{d(-t)^2} y(t) = \dfrac{d^2}{d(-t)^2} x(-t)##

Can you finish?

-Dan

## 1. How do Newton's laws demonstrate time invariance?

Newton's laws of motion state that an object will remain at rest or in motion with a constant velocity unless acted upon by an external force. This principle remains true regardless of the specific time period in which it is observed, demonstrating time invariance.

## 2. Can Newton's laws be applied to all time periods?

Yes, Newton's laws are considered to be universal and can be applied to any time period. They have been extensively tested and have been proven to hold true in a wide range of situations and time periods.

## 3. How can we test for time invariance in Newton's laws?

One way to test for time invariance is to conduct experiments in different time periods and observe if the results align with the predictions of Newton's laws. Additionally, mathematical models and simulations can also be used to analyze the time invariance of these laws.

## 4. Are there any exceptions to Newton's laws being time invariant?

There are some situations where Newton's laws may not hold true, such as at very high speeds or in the presence of extreme gravitational forces. However, these exceptions are typically accounted for by more advanced theories, such as Einstein's theory of relativity.

## 5. Why is it important to prove that Newton's laws are time invariant?

Proving the time invariance of Newton's laws is crucial in understanding and predicting the behavior of objects in motion. It allows us to make accurate predictions and calculations in a wide range of situations, making it a fundamental concept in physics and engineering.

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